Mathematical Sciences: Nonlinear Partial Differential Equations in Continuum Mechanics

小行星9600137 PI提出研究一类非线性偏微分方程 方程源于连续介质力学和自然科学的其他分支 科学，其中包括欧拉和Navier-Stokes方程， 可压缩和不可压缩，无粘和粘性流体，方程 电磁场动力学的导电可压缩 粘性流体，弹性力学方程，非线性Boltzmann型 动力学方程和多相燃烧方程 流动等。主要目标是获得动态的理解， 基本非线性波如冲击波，膨胀波， 孤立子、扩散波、涡面和边界层， 在一般流动研究中的重要作用。特别是 建议分析小尺度耗散或弛豫的影响， 大尺度物理量及其运动的过程。其他 例如强解和弱解的全局存在性问题， 基本波型的非线性稳定性，弱熵的唯一性 解，解的层性态，计算的数值方法 的不连续流动，和流体动力学限制的动力学模型也是 有待探索计划继续目前的努力， 现有的技术，并开发新的工具来研究其中的一些 问题其中一个主要的想法是改进设计的传统方法 研究光滑流或非松弛系统， 内部结构的基本不连续流，并研究这些 理论上和数值上的问题。 %%% 非线性偏微分方程的非线性演化 物理平衡定律支配着许多最常见的现象， 自然这些非线性系统的解决方案仍然是一个中心议题， 研究波的传播，是理解基本的 从可压缩和不可压缩的流体， 声波和电磁散射到波的传播 非线性复合材料、反应气体混合物、纤维光学以及磁流体力学和广义相对论。在这个项目中， 提出了研究这些非线性的一些基本问题， 系统，这是由许多领域的重要应用直接驱动的 例如图像处理、航空、复合材料、声纳、雷达 医学成像，地球物理勘探，天气预报，油藏 模拟等。例如，高分辨率设计的建议研究 混合气尖锋的多维数值计算方法 型偏微分方程是极其重要的石油 储层模拟、复合材料设计、成像处理、 和污染物的传播，如水和空气中的油;建议 边界层理论的研究是这一领域的关键内容之一。 对湍流的理解，以及超音速飞机的设计等。 因此，完成这个项目应该是相当重要的 在上述技术的改进和新开发中， 领域的 ***

9600137 Xin The PI proposes to study a class of nonlinear partial differential equations arising from Continuum Mechanics and other branches of natural sciences, which includes the Euler and Navier-Stokes equations for both compressible and incompressible, inviscid and viscous fluids, the equations of electro-magneto field dynamics for electrically conducting compressible viscous fluids, the equations of elasticity, nonlinear Boltzmann type equations in kinetic theory, and equations for combustion, multi-phase flows, etc. The main goal is to gain understanding of the dynamics and interactions of basic nonlinear waves such as shock waves, expansion waves, solitons, diffusive waves, vortex sheets, and boundary layers, which play important roles in the study of the general flows. In particular, it is proposed to analyze the effects of the small scale dissipation or relaxation processes on large scale physical quantities and their motions. Other questions such as global existence of both strong and weak solutions, nonlinear stability of basic wave patterns, uniqueness of weak entropy solutions, layer behavior of solutions, numerical methods for the computation of discontinuous flows, and fluid-dynamic limit for kinetic models are also to be explored. It is planned to continue the current effort to expand the existing techniques and to develop new tools for studying some of these problems. One of the main idea is to refine the conventional methods designed to study smooth flows or unrelaxed systems by taking into account more internal structures of the underlying discontinuous flows, and study these problems both theoretically and numerically. %%% Nonlinear evolution of partial differential equations arising from physical balance laws govern many of the most widely encountered phenomena in nature. Solutions to these nonlinear systems remain a central topic in the study of wave propagations and are fundamental to the understanding of basic physical phenomena ranging from compressible and incompressible fluids, acoustic waves and electromagnetic scattering to wave propagations in nonlinear composite materials, in reactive gas mixtures, in fiber optics, as well as in magnetohydrodynamics and general relativity. In this project, It is proposed to study some of the basic questions for these nonlinear systems, which are motivated directly by important applications to many fields such as image processing, aeronautics, composite material, sonar, radar, medical imaging, geophysical exploration, weather prediction, oil reservoir simulations etc.. For example, the proposed study of design of high resolution multi-dimensional numerical methods for calculating sharp fronts for mixed type partial differential equations is extremely important for the oil reservoir simulations, design of composite materials, imaging processing, and propagation of pollutants such as oil in water and air; the proposed study of boundary layer theory is one of the key ingredients in the understanding of turbulence, and the design of supersonic aircrafts, etc.. Thus the accomplishment of this project should be of considerable importance in the improvement and new development of technologies in the above mentioned fields. ***